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Crashes, Fat Tails, and Efficient Frontiers 2nd Edition
Crashes, Fat Tails,
and Efficient Frontiers
2nd Edition
Contents
Déjà Vu All Over Again Paul D. Kaplan
Morningstar Advisor
February/March 2009
One and a Quarter Centuries of Stock Market Drawdowns Paul D. Kaplan
Morningstar Alternative Investments Observer
Third Quarter 2009
Stock Market Bubbles and Crashes: Paul D. Kaplan
A Global Historical and Economic Perspective Morningstar Alternative Investments Observer
First Quarter 2010
Déjà Vu All Around the World Paul D. Kaplan
Morningstar Institutional Perspective
October 2009
Nailing Downside Risk James X. Xiong
Morningstar Advisor
February/March 2010
Getting a Read on Risk Paul D. Kaplan, Roger Ibbotson,
George Cooper, Benoit Mandelbrot
Morningstar Advisor
February/March 2009
MPT Put Through the Wringer Paul D. Kaplan, Steven Fox, Michael Falk
Morningstar Advisor
August/September 2009
Markowitz 2.0 Paul D. Kaplan, Sam Savage
Morningstar Advisor
April/May 2010
© 2010 Morningstar. All rights reserved. The information contained herein: (1) is intended solely for informational purposes; (2) is proprietary to Morningstar and/or its content providers; (3) may not
be copied or distributed; (4) is not warranted to be accurate, complete, or timely; and (5) does not constitute investment advice of any kind. Neither Morningstar nor its content providers are responsible for
any damages or losses arising from any use of this information. Past performance is no guarantee of future results. “Morningstar” and the Morningstar logo are registered marks of Morningstar, Inc.
Spotlight
Déjà Vu All Over Again
By Paul D. Kaplan
When risk models fall short, advisors need to look no further than
the historical record to plan for the next 100-year flood.
“We seem to have a once-in-a-lifetime crisis confidence in equity markets, and avoid a deep Take, for example, the poster’s depiction of the
every three or four years.” global recession. compound annual return of the S&P 500
—Leslie Rahl, founder of Capital Market Index, identified on the chart as Large Stocks.2, 3
Risk Advisors1 If you need to be reminded how bad things are, The growth of $1 to $2,049 over 83 years
listen to our political and fiscal-policy leaders is impressive (a rate of 9.6% per year),
The dramatic events on Wall Street and in as they describe the crisis with phrases that but the record is peppered with several long
financial centers around the world that started begin with the ominous words “once in a … .” and severe declines, some in the not-too-
on “Black Sunday,” Sept. 14, have upset As they were pushing their $700-billion bailout distant past.
many common assumptions about the global package last fall, members of the Bush
financial system. What started as a mortgage administration said that the crisis was a To illustrate our point, we isolated the S&P 500
crisis spread to nearly every corner of “once-in-a-century event,” and this was echoed line of the poster and added blue areas
the financial system when Lehman Brothers in November by Henry Paulson, the former that show the highest level that the cumulative
collapsed, Merrill Lynch sold itself to Bank secretary of the U.S. Treasury, who said the value of the S&P 500 had achieved as of
of America, and AIG became strapped for meltdown was a “once- or twice-in-a-100-year that date (Exhibit 1). Wherever a blue area is
cash—all in a single weekend. These and the event.” Former Federal Reserve chairman shown, the S&P 500 was amid a decline
events that followed have shaken investor Alan Greenspan characterized the crisis as a relative to its most recent peak. The deeper the
confidence to the core. As of Dec. 31, the Dow “once-in-a-century credit tsunami.” gap, the more severe the decline; the wider
Jones Industrial Average was down 22.4% the gap, the longer the time until the S&P 500
since Black Sunday. The yield spread on junk There’s little doubt that aspects of this crisis returned to its peak. Wherever a blue area
bonds over LIBOR reached an unprecedented are unique and that the economy is facing its is not shown, the S&P 500 was climbing to a
16%. The markets for many assets have hardest challenge since the Great Depression, new peak.
become illiquid, and credit is dried up for nearly but are severe economic crises the rare events
anyone who needs it. The U.S. Federal Reserve, Paulson, Greenspan, et al., have suggested? Not surprisingly, the granddaddy of all market
the U.S. Treasury, and their counterparts A study of capital market history suggests no. declines started with the Crash of 1929 and did
around the world have taken dramatic steps to To see this, you need to look no further than the not recover until 1945. The S&P 500 lost more
restore liquidity to asset markets, stimulate Ibbotson Stocks, Bonds, Bills, and Inflation than 83% of its value in about three years
lenders to make loans again, shore up investor poster from Morningstar hanging on your wall. and took 121/2 years to recover. What may be
1 As quoted by Christopher Wright, “Tail Tales,” CFA Institute Magazine, March/April 2007. 2 We obtained the historical monthly total returns from Morningstar EnCorr, an institu-
tional asset-allocation software and data package. 3 We use a logarithmic scale for all growth of $1 charts.
28 Morningstar Advisor February/March 2009
Exhibit 1: Mind the Gaps U.S. large-cap stocks have made impressive gains over the years, but several significant
declines have interrupted the S&P 500’s trajectory.
10,000 Growth of $1 invested in S&P 500 Highest cumulative level of S&P 500 as of date point
Crash of 1987
1,000 Dot-com
bubble burst
1969 recession
100
Arab oil embargo
Crash of 1929 and
Great Depression 1962 bear market
10
Post-war
1 manufacturing crisis
1925 1930 1935 1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005
Growth of $1 includes reinvested dividends. Monthly data used to calculate returns.
more sobering, however, is that the second- Measuring Risk: The Standard Model throughout the investment profession. The
greatest decline took place within the past With 20% declines occurring, on average, best known of these models are the capital
decade. With the crash of the Internet bubble every decade or so, you’d think that the asset pricing model of expected returns
in 2000, the S&P 500 lost almost 45% of standard risk models that investors use to and the Black-Scholes option pricing model.
its value over a two-year period and took four make their asset-allocation decisions would These models’ creators have won the
years to return to its peak value. assign a significant probability that these Nobel Prize in economics for their path-break-
events will occur. Think again. To see why, ing work. Each of these models starts by
In all, including the current crisis, the S&P we need to look at the history of how these making an assumption about the statistical
500 has suffered eight peak-to-trough declines models were formed. distribution of stock market returns. The
of more than 20% since the mid-1920s. CAPM assumes that returns follow a normal,
Two of the three greatest declines occurred To help make sense of the highly complex or bell-shaped, distribution. The Black-
in the past eight years. To suggest that capital markets, financial economists in 1960s Scholes model assumes that returns follow a
the current crisis is a once-in-a-century event and 1970s developed a set of mathematical lognormal distribution.4
ignores the record. models of the markets that are used to this day
4 For returns to follow a lognormal distribution means that logarithm one plus the return in decimal follows a normal distribution.
MorningstarAdvisor.com 29
Spotlight
Exhibit 2: Cracks in the Bell Standard risk models assume S&P 500 returns follow a bell-shaped distribution, even
though the index has experienced more than 10 declines of at least –13%.
200
180 10
160
140 5
120
100
–29% –25% –21% –17% –13%
80
60
40
20
–29% –21% –13% –5% 3% 11% 19% 27% 35% 43%
Histogram shows the frequency of monthly returns for the S&P 500 from January 1926 to November 2008.
With these standard models, the primary We can illustrate the problem further by Mandelbrot’s model to stock prices and
measure of risk is standard deviation. If returns overlaying a lognormal model of returns over obtained promising results.5 Until recently,
follow a normal distribution, the chance a histogram of monthly total returns on however, the work of Mandelbrot and Fama
that a return would be more than three the S&P 500 (Exhibit 2). The model says that had been largely ignored.6
standard deviations below average would be declines of more than negative 13% have
a trivial 0.135%. Since January 1926, we almost no chance of happening—yet they have In his dissertation, Fama assumed that the
have 996 months of stock market data; 0.135% occurred at least 10 times since 1926. logarithm of stock returns followed a fat-tailed
of 996 is 1.34—that is, there should be distribution called a “stable Paretian distribu-
only one or two occurrences of such event. An Alternative Approach: Log-Stable tion,” or stable distribution.7 Hence, we refer to
Distributions the resulting distribution of returns as a
But the record of the stock market tells a In the early 1960s, Benoit Mandelbrot, a “log-stable distribution.”
different story. The monthly returns of the S&P mathematician teaching economics at the
500 have been more than three standard University of Chicago, was advising a doctoral We can illustrate an example of Fama’s
deviations below average 10 times since 1926. student named Eugene Fama. Mandelbrot work by using the same S&P 500 histogram
In other words, the standard models assign had developed a statistical model for percent- in our earlier exhibit but with a log-stable
meaninglessly small probabilities to extreme age changes in the price of cotton that had distribution curve overlaying it instead of a
events that occur five to 10 times more than “fat tails.” That is, the model assigned nontrivial lognormal curve.8 The log-stable model
the models predict. probabilities to large percentage changes. (Exhibit 3) fits the empirical distribution much
In his doctoral dissertation, Fama applied closer than the lognormal both at the
5 For an account of the work of Mandelbrot and Fama during this period, see Benoit Mandelbrot and Richard L. Hudson, The (Mis)Behavior of Markets, New York: Basic Books,
2004. 6 The idea of using fat-tailed distributions to model asset returns is starting to gain some traction. FinAnalytica was founded to provide investment analysis and portfolio
construction software based on Mandelbrot and Fama’s work. Morningstar added distribution charts and forecasting models based on it to Morningstar EnCorr. 7 Strictly speaking,
the assumption is that the logarithm of one plus the return in decimal form follows a stable Paretian distribution. 8 This chart can be produced in Morningstar EnCorr Analyzer
using the log-stable feature.
30 Morningstar Advisor February/March 2009
Exhibit 3: It’s a Fat-Tailed World, After All A log-stable distribution does a good job of modeling the empirical
returns of the S&P 500, especially at the center and the tails.
200
180 10
160
140 5
120
100
–29% –25% –21% –17% –13%
13
80
60
40
20
–29% –21% –13% –5% 3% 11% 19% 27% 35% 43%
Histogram shows the frequency of monthly returns for the S&P 500 from January 1926 to November 2008.
center and the tails. In particular, note the distributions, the normal or bell-shaped
Exhibit 4
close match between the density curve and the distribution being the best-known example.
Power Law Tails: Unlike a normal distribution, a
histogram between negative 13% and Other distributions assign so much probability
stable distribution approaches the straight line of
negative 29%. to the tails that variance is infinite. Such is a power law, indicating that it has “fat tails.”
the case with stable distributions.
The tails of a stable distribution are so fat that In (Prob[X
Spotlight
loss for a normal distribution, a stable
distribution, and a power law distribution. The
line for the normal distribution curves down, Hard Eight
indicating that it has thin tails. In contrast, the
line for stable distribution approaches the The S&P 500 has suffered eight peak-to-trough declines of more than 20%.
straight line of the power law because it is very
similar to a power law for large losses. Peak Trough Decline % Recovery
August 1929 June 1932 83.41 January 1945
These results show that the log-stable August 2000 September 2002 44.73 October 2006
distribution does a good job of modeling the December 1972 September 1974 42.64 June 1976
empirical returns distribution of the S&P 500. October 2007 November 2008 40.89 To Be Determined
The better fit of the log-stable distribution August 1987 November 1987 29.58 May 1989
demonstrates that the S&P 500 has fatter tails November 1968 June 1970 29.16 March 1971
than predicted by the lognormal model. It December 1961 June 1962 22.28 April 1963
also calls into question commonly used May 1946 November 1946 21.76 October 1949
portfolio construction techniques such as the
Table shows the worst cumulative peak-to-trough declines in percentage terms since December 1925. Based on monthly returns.
mean-variance optimization, which relies
on the assumption of a finite variance.
If the log-stable model does such a better job
in describing the distribution of asset returns, random variables are infinite. The lack of a an investment, advisors would benefit by
why has it not received more acceptance? finite variance means that most portfolio beginning to think about a more complete
There are several possible reasons. First, the theories and most portfolio construction risk model. A complete risk model allows
mathematics is challenging. Second, the techniques are invalid, including those based investors to consider three questions about a
variances and all higher moments of stable on alternative risk measures such as “down- potential decline in value simultaneously:
side risk.” Finally, there is no single obvious
Exhibit 5 way to estimate the parameters of stable distri- r How likely might a decline occur?
Role of Time: The log-stable model indicates butions as there is with normal distributions. r How long might it last?
that there’s a 4% to 5% probability that the S&P r How bad might it get?
500 will lose 50% or more over extended time
Risk Measures versus Risk Models
periods. The lognormal model puts the odds much
For advisors, the lesson here is not that they It is already common practice in some
lower.
should throw away the standard ways of segments of the financial-services industry
Probability of Drop of 50% or More summarizing risk using measures such as to use a risk model to measure “value at
6%
standard deviation and downside deviation.10 risk”—that is, how bad a loss might be
Log-Stable
g Nor should advisors run to embrace Fama’s over a given length of time and with a given
log-stable models. probability.
4%
Instead, we think advisors should understand As you can appreciate through our study of
the limitations of standard risk measures and historical stock market declines, time horizon
2%
have a basic understanding of what Mandel- is a key dimension of risk not explicitly
Lognormal
brot’s and Fama’s work says about describing addressed by standard risk measures. A
5 10 15 20 25 30 35 40 45 50 risk. Rather than solely relying on a few complete risk model can be used to explicitly
Number of Years summary statistics to characterize the risks of take time horizons into account.
10 In recognition that return distributions may not be symmetric, measures such as skewness and kurtosis are sometimes presented alongside standard deviation. However, like
variance, these measures are not defined for stable Paretian distributions.
32 Morningstar Advisor February/March 2009For example, in Exhibit 5, we plot the
probability of a cumulative loss of 50% or more We Are Not Alone
over various time horizons using the lognormal
distribution for the S&P 500 that we show The uneven performance of the stock market is hardly unique to the United States. Severe
in Exhibit 2 and the log-stable distribution in declines—mostly within the past decade—have occurred in developed markets since January
Exhibit 3. The lognormal model shows that the 1970. Here are the worst declines for seven countries.
risk of such a severe decline over an extended
period is negligible. The log-stable model, Country Peak Trough Decline % Recovery
on the other hand, indicates that such a loss Germany February 2000 March 2003 67.89 April 2007
over an extended period has a probability of Japan December 1989 April 2003 67.62 To Be Determined
4% to 5%—numbers significant enough to gain U.K. August 1972 November 1974 64.73 January 1977
the attention of risk-averse advisors and Italy June 1973 December 1977 59.39 September 1980
investors who might want to be prepared for Spain April 1974 November 1979 58.81 March 1984
such a scenario. France August 2000 March 2003 58.28 March 2007
Canada August 2000 September 2002 47.11 September 2005
Conclusion Source: Morgan Stanley Capital International and Morningstar EnCorr. Chart shows monthly return data in local currency for major
In every financial crisis, investors relearn stock-market index in each country.
the same message—there isn’t a magic risk
measure or model that can account for
or predict every significant drop in the market. The Japanese market has yet to recover from its peak in December 1989.
Economists and quantitative analysts have
100
made incredible strides over the decades
engineering new ways to explain the distribu-
tion of returns. These developments provide
investors with valuable information to help 10
them decide how to allocate their portfolios
for any number of investing scenarios
and mitigate risk. But they are not perfect. 1
As we’ve shown, the record contains a much
bumpier ride than many risk models would 1970 1975 1980 1985 1990 1995 2000 2005
suggest. In addition to preparing clients’
portfolios for these occasional severe declines
and taking other precautions, advisors would The markets in four of the seven countries have performed worse since October 2007 than the U.S.
do well to keep reminding their clients of market, which has fallen 40%.
the risks they face as investors. Clients should
50% Italy Japan Germany France Spain Canada U.K.
be fully prepared0to take on the 100-year
Decline % –48.69 –47.44 –43.04 –42.14 –39.42 –34.85 –31.26
floods they will
-10
surely face in the future. K
Paul D. Kaplan,-20
Ph.D., CFA, is Morningstar’s vice presi-
dent of quantitative research and a 30
frequent contributor
-30
to Morningstar Advisor.
-40
-50
10 Data through December 2008. Based on monthly returns.
MorningstarAdvisor.com 335 Morningstar Alternative Investments Observer
Third Quarter 2009
Quant Corner:
One and a Quarter Centuries
of Stock Market Drawdowns
Real stock market returns reveal the
true frequency of “once-in-a-century” crashes.
available in its EnCorr® software and data to 1871. Unfortunately, Professor Shiller’s
package that starts in 1926. The results clearly stock data is based on monthly average prices
demonstrate that Greenspan was in need of a rather than month-end prices. So I could
history lesson. use his inflation data, but not his stock market
data. Separately, Roger Ibbotson and some
by
Paul D. Kaplan, Ph.D., CFA I have recently expanded the analysis into a colleagues created an annual price and total
Vice President, complete study on global equity market history return series for the NYSE that goes back
Quantitative Research
upon the request of Larry Siegel, director of to 1825.5 However, annual returns are at too
research at the CFA Institute, as a contribution low a frequency to measure the largest
to his forthcoming book on the global history drawdowns of the period, such as the large
When former Federal Reserve chairman Alan of market crashes.3 Larry asked me to use drop in the stock market during the panic of
Greenspan characterized the financial monthly real total returns4 and to go back into 1907. Fortunately, Larry had a book that
crisis of 2008 as a “once-in-a-century credit history as far as it was possible with reason- contained daily price data on the Dow Jones
tsunami,” I was stunned. Being familiar ably reliable data. The benefit of using real Averages going back to 1885.6 He advised
with long-term data on the U.S. capital returns is to make meaningful return compari- me to estimate the monthly price returns in the
markets, I thought a more apropos statement sons, as our study spans such a long period of broader NYSE price index from the monthly
was the one made by Leslie Rahl (founder time. The benefit of going further back in price returns on the Dow Jones Averages and
of Capital Market Risk Advisors) more than year history is, of course, to give us a longer-term then interpolate the total returns by assuming
before the crisis when she said, “We seem and more robust historical perspective on that the level dividends remained constant
to have a once-in-a-lifetime crisis every three market crashes, in terms of frequency, length, during each year.
or four years.”1 The contrast between and magnitude.
Greenspan’s and Rahl’s perspectives inspired Following Larry’s advice, and soliciting the
me to write an article for Morningstar To complete the study, I needed to find monthly help of Morningstar intern Kailin Liu,
Advisor on the history of market meltdowns, data from before 1925 on both stock returns I produced a time series of real total returns for
“Déjà Vu All Over Again.”2 In that article, and inflation, and calculate real returns. Since the U.S. stock markets that runs from 1871
I illustrate the frequency and severity of the there was no such return series in existence, through the present. While for the first 15 years
major drawdown for various countries I would have to create one out of readable we only have annual returns, we now have
using time series of stock market total returns. available data. more than 123 years of monthly total real
For the U.S., I naturally used the series returns. This data will appear in future editions
on the S&P 500 that Morningstar publishes in Professor Robert Shiller of Yale posts a of the Ibbotson SBBI Yearbooks, beginning
the Ibbotson® SBBI® Yearbooks and makes monthly history of U.S. stock market returns in 2010.
and inflation on his Web site that goes back C ON T I N UE D ON N E X T PAGEOne and a Quarter Centuries of Stock Market Drawdowns continued 6 Morningstar Alternative Investments Observer
Third Quarter 2009
Exhibit 1: Real Index and Peak Values of the U.S. Stock Market
10,000
Real Cumulative Value
Dot-Com Bubble Burst
Crash of 2007–09
1,000
Crash of 1987
Inflationary Bear Market of 1973–74
100
Postwar Bear Market
Great Depression
10
Panic of 1907
Auto Bubble Burst
$1
1870 1880 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010
Exhibit 2: Largest Declines in U.S. Stock Market History (in real total return terms, from January 1871 to June 2009)
Peak Trough Decline % Recovery Event(s)
August 1929 May 1932 79.00 November 1936 Crash of 1929, 1st part of Great Depression
August 2000 February 2009 54.00 TBD Dot-Com Bubble Burst (2000–02), Crash of 2007–09
December 1972 September 1974 51.86 December 1984 Inflationary Bear Market, Vietnam, Watergate
June 1911 December 1920 50.96 December 1924 World War I, Postwar Auto Bubble Burst
February 1937 March 1938 49.93 February 1945 2nd part of Great Depression, World War II
May 1946 February 1948 37.18 October 1950 Postwar Bear Market
November 1968 June 1970 35.46 November 1972 Start of Inflationary Bear Market
January 1906 October 1907 34.22 August 1908 Panic of 1907
April 1899 June 1900 30.41 March 1901 Cornering of Northern Pacific Stock
August 1987 November 1987 30.16 July 1989 Black Monday—Oct. 19, 1987
October 1892 July 1893 27.32 March 1894 Silver Agitation
December 1961 June 1962 22.80 April 1963 Height of the Cold War, Cuban Missile Crisis
November 1886 March 1888 22.04 May 1889 Depression, Railroad Strikes
April 1903 September 1903 21.67 November 1904 Rich Man’s Panic
August 1897 March 1898 21.13 August 1898 Outbreak of Boer War
September 1909 July 1910 20.55 February 1911 Enforcement of the Sherman Anti-Trust Act
May 1890 July 1891 20.11 February 1892 Baring Brothers CrisisOne and a Quarter Centuries of Stock Market Drawdowns continued 7 Morningstar Alternative Investments Observer
Third Quarter 2009
Truth in Numbers The history of stock market drawdowns
The significance of this data is in the lessons presented here shows that investing in stocks
that we can learn from it. Over the entire can be very risky business, indeed, and
138½-year period, the Real US Stock Market that the current crisis is hardly a “once-in-a-
Index grew from $1 to $5,179 in 1869 dollars. century” event. But to more than just state
This is a compound annual real total of just the obvious, we should use this data to better
under 6.4%, almost the same as the post-1925 gauge the potential risks and long-term
period. However, as Exhibit 1 shows, it rewards of investing in risky assets such as
was a very bumpy ride with a number of major stocks. Specifically, we should supplement
drawdowns, some of which can be linked our traditional measures of risk, such as
with specific economic and political events. standard deviation, which relies on a normal
distribution, by measures that better
Exhibit 1 shows the growth of $1 invested in capture the fat-tailed nature of the historical
the U.S. stock market at the end of 1869 returns and drawdowns as presented here.
through June 2009 in real terms, along with a Incorporating fat-tailed distributions
line that shows the highest level that the into risk measures has become a focus of my
index had achieved as of that date. Wherever research. Stay tuned for more. K
this line is above the cumulative value line,
the index was amid a decline relative to its
most recent peak. The bigger the gap, the more References
severe the decline; the wider the gap, the 1 As quoted by Christopher Wright, “Tail Tales,”
longer the time until the index returned to its CFA Institute Magazine, March/April 2007.
peak. Wherever this line coincides with 2 February/March 2009. Available at
the index line, the index was climbing to a http://www.nxtbook.com/nxtbooks/morningstar/
new peak. advisor_20090203/.
3 This study will appear in the CFA Institute’s
Exhibit 2 lists all of the drawdowns that forthcoming book, Voices of Wisdom:
Understanding the Global Financial Crisis,
exceeded 20%. In total, there were 17 such
Laurence B. Siegel, editor.
declines, including the present one from
4 That is, returns that include the reinvestment
which we have yet to recover. Not surprisingly, of dividends and are adjusted for inflation.
the granddaddy of all market declines
5 Goetzmann, William N., Roger G. Ibbotson,
started just before the Crash of 1929 and did and Liang Peng, “A New Historical Database for
not recover until toward the end of 1936. the NYSE 1815 to 1925: Performance and
The U.S. stock market lost 79% of its real value Predictability,” Journal of Financial Markets,
in less than three years, and it took more December 2000. The data appear in the Ibbotson
SBBI Yearbooks.
than five years to recover. What may be more
6 Pierce, Phyllis, ed. 1982. The Dow Jones Averages
sobering, however, is that not only are
1885–1980. Dow Jones Irwin, Homewood, Illinois.
we currently in the second-greatest decline,
but it started nine years ago! The combined
effect of the crash of the Internet bubble
in 2000 and the financial crisis of 2008 caused
the U.S. stock market to lose 54% of its real
value from August 2000 to February 2009.
Who knows how long it will take to recover
from that and when our next crisis will occur?5 Morningstar Alternative Investments Observer
First Quarter 2010
Quant Corner: Stock Market
Bubbles and Crashes
A global historical and economic perspective.
not threaten to impair the real economy, its present and compare them with the indexes
production, jobs, and price stability.’ for Japan and the United States over that same
period to see which of the more recent
“Immediately after [Greenspan] said this, the crashes were regional and which were global
stock market in Tokyo, which was open as in nature. Finally, we look to economic theory to
by
Paul D. Kaplan, Ph.D., CFA he gave this speech, fell sharply, and closed help explain bubbles and crashes and apply
Quantitative Research down 3%. Hong Kong fell 3%. Then markets in these theories to the recent financial crisis.
Director,
Morningstar Europe
Frankfurt and London fell 4%. The stock market While we don’t think bubbles and crashes can
in the U.S. fell 2% at the open of trade.” be prevented entirely, we believe that
necessary steps must be taken to reduce the
Adapted from “The History and Economics of Stock Although it is unlikely that Greenspan’s simple frequency and magnitude of financial crises.
Market Crashes,” by Paul D. Kaplan, Thomas Idzorek,
statement was intended to cause the reaction
Michele Gambera, Katsunari Yamaguchi,
James Xiong, and David M. Blanchett, in Insights into that it did, the term “irrational exuberance” The U.S. Record
the Global Financial Crisis, Laurence B. Siegel, ed., has now become associated with any period Kaplan (2009) presents the real total return
©2009 Research Foundation of CFA Institute. Portions when investors are in a heightened state index and the peak values of the U.S.
reproduced and republished with permission of speculative fervor. Speculative fervors, or stock market over the period January 1871
from the Research Foundation of CFA Institute.
All rights reserved.
bubbles as they are more popularly known, through June 2009, a period of just more
may be easy to identify with the benefit of hind- than 138 years. (See Morningstar Alternative
sight, but they are not nearly as easy to Investments Observer, Third Quarter 2009.)
According to Shiller (2005), the term “irrational identify when they are occurring. Moreover, Kaplan shows that an investment in a
exuberance” is credited to Alan Greenspan, they are not by any means new phenomena. hypothetical index fund of the U.S. stock market
former chairman of the U.S. Federal Reserve Even though the recent market crash beginning held over this period (with all dividends
Board. In his book, Irrational Exuberance, Shiller in 2007 is likely fresh on the mind of the reinvested and no taxes, fees, or other costs)
explains that Greenspan used this term in a reader, there have been many others, all around would have grown nearly 5,000-fold in
1996 speech: the world, and some far worse. real purchasing power. Nonetheless, a number
of significant sharp and/or long declines
“ ‘But how do we know when irrational exuber- To place the market meltdown of 2008–2009 in occurred along the way. The periods where
ance has unduly escalated asset values, which historical perspective, we examine the there are gaps between the peak and the
then become subject to unexpected and long-term record of stock market total return index are the times—called “drawdowns”—
prolonged contractions as they have in Japan indexes1 for the United States, the United when the market in question fell below its own
over the past decade?’ [Greenspan] added that, Kingdom, and Japan. We also examine the immediate past peak and later recovered.
record of the regional stock market indexes C ON T I N UE D ON N E X T PAGE
‘We as central bankers need not be concerned (stated in U.S. dollars) for Asia ex-Japan,
if a collapsing financial asset bubble does Europe, and Latin America from 1988 to the 1 Total Return Indexes include reinvestment of dividends.Quant Corner: Stock Market Bubbles and Crashes continued 6 Morningstar Alternative Investments Observer
First Quarter 2010
The U.K. Record regain the peak reached in April 1972 until with the stock market peaking in December
The long-term equity returns for the United
1,000
January 1984, roughly 12 years later. 1989. The compound annual real total
Kingdom bear a striking resemblance to return of the Tokyo Stock Price Index, or TOPIX,
those of the United States, highlighting how The 74 percent drawdown in the real total from January 1952 to December 1989 was
connected the two economies have been. return index of U.K. stocks in the 1970s is much 13.4 percent.3 The market then declined
100
The largest shock to the U.K. stock market over worse than that same market’s decline for much of the subsequent two decades—with
the past 109 years occurred shortly after in the Great Depression, despite the much more stock prices falling 71.9 percent from
the collapse of the Bretton Woods system and severe damage to the real economy in the the 1989 peak, in real terms, by March 2009.
during the oil crisis that began Oct. 17, 1973, earlier episode. Thus, markets do not always Exhibit 2 includes information on the major
10
when members of the Organization of track real economic events exactly or declines in the Japanese stock market during
Arab Petroleum Exporting Countries, or OAPEC, even somewhat closely, as shown in Exhibit 1. the past six decades.
proclaimed an oil embargo against select
1
industrial governments of the world to pressure Japanese Record It is important to distinguish between market
1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010
Israel during the fourth Arab-Israeli War.2 The Japanese economy experienced a strong declines caused by business cycles and
Although the embargo was officially lifted in recovery following World War II and had those caused by sudden unexpected crashes in
March 1974, the U.K. stock market did not relatively consistent growth through the 1980s, Japan, as well as in other markets.
C ON T I N UE D ON N E X T PAGE
Exhibit 1: U.K. Stock Market History, 1900–2009
1,000
Peak Trough Decline % Recovery
Apr-72 Nov-74 73.81 Jan-84
1913 1920 45.85 1922
100 Dec-99 Jan-03 44.91 Apr-07
1936 1940 43.71 1946
Oct-07 Feb-09 40.99 TBD
10
1968 May-70 35.80 Apr-72
Sep-87 Nov-87 34.07 Nov-92
1928 1931 30.57 1933
1946 1952 21.30 1954
1 Jan-94 Jun-94 17.11 Nov-95
0.1
1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010
Exhibit 2: Japanese Stock Market History, 1952–2009
1,000
Peak Trough Decline % Recovery
Dec-89 Mar-03 71.92 TBD
Dec-72 Oct-74 51.85 Dec-83
Jun-61 Jun-65 34.47 Aug-68
100 Jan-53 May-54 31.98 Dec-55
Mar-70 Dec-70 21.33 Jun-71
Aug-87 Dec-87 19.79 Mar-88
Mar-57 Jul-57 17.77 Jun-58
Jul-71 Oct-71 15.58 Jan-72
10 Mar-84 Jul-84 12.66 Dec-84
Apr-60 May-60 11.62 Aug-60
1
1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010
2 The now better-known OPEC, Organization of Petroleum Exporting Countries, is a separate, overlapping organization.
3 The Tokyo Stock Exchange, TSE, is divided into three markets: the first section, the second section, and Mothers (venture capital market). The first section includes the largest, most successful companies.
The TOPIX tracks all domestic companies of the TSE’s first section. See www.tse.or.jp/english/faq/list/general/g_b.html.Quant Corner: Stock Market Bubbles and Crashes continued 7 Morningstar Alternative Investments Observer
First Quarter 2010
For example, the decline that began in economy. As a result, the health of the financial John Maynard Keynes (1936) set forth a theory
December 1972 was triggered by currency sector is a key factor in the economic cycle. that markedly differed from those of his
instability and rising interest rates following At the same time, economic theory has predecessors. He argued, loosely speaking, that
the first oil crisis. The 1961–65 decline devoted increasing attention to the causes of some special markets are almost never
was caused, at first, by a tightening of financial crises. in equilibrium, For example, the labor market is
monetary policy and deteriorating corporate generally in disequilibrium. Financial markets,
earnings, culminating in a financial market Economic Thought and Financial Crises Keynes quipped, “can stay irrational longer
crisis that led to a bailout of Yamaichi Adam Smith stated that the existence of many than you can stay solvent.” With this, he meant
Securities in 1965. Those are bear markets— small banks is a guarantee for the public that financial markets are not perfectly efficient
continuous declines caused by changes in because, among other things, it limits the and that government policy, specifically fiscal
fundamentals but without a big one-day or systemic effect of the failure of any one bank “stimulus” (deficit spending to accelerate the
several-day “crash.” (Smith 1776, Book II, Chapter II). Apart from demand for goods and services), may be
Smith’s remarks, bank size was not at a necessary remedy when a serious recession
Drawdowns During the Long Boom the heart of economic theory until recently. ensues. Not everybody knows that Keynes
(1982–2007) The banks at the core of the recent crisis are did not advocate large, persistent government
Stock markets around the world have very large ones. If Smith’s observation is budget deficits; he supported only focused
experienced a number of large drawdowns over accurate, then something must be wrong with actions against the most serious recessions.
the past 20 years. Most of the period from very large banks.
January 1988 to June 2009 marked a time Hyman P. Minsky (1986, 1992) studied why
frame of continued growth for many countries Joseph Schumpeter (1942) brought a new markets are, in Keynes words, irrational,
and stock markets, a period often characterized perspective to economic theory related whereas Modern Portfolio Theory relied heavily
as the “Long Boom.” Drawdowns of more to financial crisis, although his views were not on market efficiency, which is the exact
than 50 percent, however, have actually intended as an explanation of one. In his contrary. Minsky’s insights fit nicely with the
occurred relatively frequently, even during the view, technical innovation causes short-term findings of behavioral finance. Briefly,
Long Boom. Generally, they have occurred in disequilibrium in markets, and that such Minsky argued that a lack of crises is the cause
emerging-markets nations. disequilibrium is a good thing because it fosters of future crises; that is, market stability
product variety and technical efficiency. is self-destructing. When market participants
Apart from the crash of 2007–2009, both the Moreover, disequilibrium would be limited only have been in a state of calm, they start
Asia ex-Japan and Latin America stock to the markets where an innovation has believing that markets will remain calm for the
markets have experienced market declines recently occurred. foreseeable future and, therefore, start
(in some cases experienced as crashes) underestimating risk. As a result, they behave
of more than 50 percent. Exhibit 3 (Page 8) J.G. Knut Wicksell and Irving Fisher (see, for just like the overoptimistic bankers of Wicksell
includes information on drawdowns around the example, Fisher 1933) introduced a view of and Fisher. Minsky suggested some government
world in various markets from January disequilibrium that specifically centered on intervention to prevent this kind of excess.
1988 to June 2009. Unfortunately, we do not financial markets, particularly the difference
have data covering emerging markets in between the market interest rate and the Finally, Friedrich A. Hayek (1932) believed that
the first years of the Long Boom, 1982–1987. equilibrium interest rate. The Walrasian model government intervention actually triggers
shows that, in a competitive equilibrium, the a Wicksell-Hayek crash, in which the market
Why Do Crashes Occur? interest rate should equal the marginal interest rate diverges from the natural rate.
Financial crises and bank failures have occurred productivity of capital.4 But Wicksell and His view was that when governments and
throughout history. As an example, Calomiris Fisher pointed to a situation where the market central banks try to expand credit to sustain the
(2008) mentions a bank panic in ancient Rome interest rate differs from the equilibrium economy when a recession is feared, as
in A.D. 33. In economies where subsistence interest rate. The theory presented by Wicksell they typically do, they end up causing a deeper
farming and barter were widespread, however, and Fisher implies that excessive lending recession. Hayek trusted markets to be efficient
banking crises affected only a small part of the causes financial crises that can stop an entire enough to take care of themselves; prices
population. In today’s world, banks and economy because they cause first a bubble and and wages would change, and markets would
insurance companies affect a large part of the then a crash in many markets at the same time. C ON T I N UE D ON P. 9
4 Financial practitioners may be a bit puzzled here because most economic theory relies on just one interest rate, with neither a yield curve (because models often focus on one or two periods) nor a credit spread
(because there is no uncertainty). If that is your point of reference, please bear with us because there are useful insights for everyone in the finance viewpoint, which incorporates multiple time horizons and uncertainty.Quant Corner: Stock Market Bubbles and Crashes continued 8 Morningstar Alternative Investments Observer
First Quarter 2010
Exhibit 3: Worst Drawdowns Around the World, January 1988–June 2009 (U.S. Dollars)
Asia ex-Japan Peak Trough Decline % Recovery
Dec-93 Aug-98 64.55 Dec-05
Oct-07 Feb-09 61.50 TBD
Jul-90 Sep-90 27.30 Dec-91
Apr-89 Jun-89 11.25 Sep-89
Jul-88 Aug-88 8.30 Dec-88
Apr-06 Jun-06 7.78 Oct-06
Oct-92 Dec-92 6.46 Feb-93
Jun-92 Aug-92 5.77 Oct-92
Mar-90 Apr-90 4.08 May-90
Japan
May-93 Jun-93 2.61 Aug-93
Dec-89 Mar-03 62.81 TBD
Feb-89 Jun-89 11.38 Sep-89
Apr-88 Aug-88 11.00 Nov-88
Sep-89 Oct-89 2.68 Nov-89
Europe
Oct-07 Feb-09 59.78 TBD
Mar-00 Sep-02 45.73 Dec-04
Jul-90 Sep-90 20.49 May-92
Jul-98 Sep-98 16.50 Apr-99
May-92 Nov-92 13.62 Aug-93
Jan-94 Jun-94 7.41 Aug-94
Dec-99 Jan-00 7.00 Mar-00
Apr-88 Aug-88 6.91 Oct-88
Sep-89 Oct-89 6.50 Dec-89
Latin America
Jul-97 Aug-97 5.62 Sep-97
May-08 Feb-09 61.12 TBD
Jul-97 Aug-98 51.34 Dec-03
Sep-94 Mar-95 42.23 Apr-97
Feb-90 Mar-90 30.80 Jul-90
May-89 Jun-89 25.00 Feb-90
May-92 Sep-92 24.85 Aug-93
Jul-90 Oct-90 20.22 Feb-91
Jan-94 Jun-94 17.12 Aug-94
Apr-06 May-06 13.89 Oct-06
United States
Mar-04 May-04 11.08 Sep-04
Oct-07 Feb-09 50.95 TBD
Aug-00 Sep-02 44.73 Oct-06
Jun-98 Aug-98 15.37 Nov-98
May-90 Oct-90 14.70 Feb-91
Jan-94 Mar-94 6.93 Aug-94
Dec-99 Feb-00 6.82 Mar-00
Dec-89 Jan-90 6.71 May-90
Jun-99 Sep-99 6.24 Nov-99
Jul-97 Aug-97 5.56 Nov-97
Mar-00 May-00 5.00 Aug-00
1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010Quant Corner: Stock Market Bubbles and Crashes continued 9 Morningstar Alternative Investments Observer
First Quarter 2010
go back to equilibrium right away. He thought to purchase higher-yielding mortgage-backed should consider this kind of risk when building
that workers should accept lower wages securities, without too much worry about portfolios and developing their risk models.
when the marginal product of their labor the quality of the securities and, therefore, the Moreover, we believe that some aspects
decreased and that governments prevented sustainability of the yields. Bond-rating of the financial infrastructure, such as the
wage falls for demagogic reasons, which in agencies, whose income (ironically) comes from derivatives market, need reform. In particular,
the end hurt workers. bond issuers, made billions of dollars a reduction of over-the-counter derivatives
by trusting faulty risk models that gave AAA transactions would lead to a more transparent
2007–09 Crash ratings to questionable mortgage-backed and safe financial sector. K
How do the events of 2007–2009 fit into the securities. Regulators did not recognize the risk
aforementioned theories? It is now clear of excessive leverage and allowed banks References
that many financial institutions had taken on and other nondepository financial firms— Calomiris, Charles W. 2008. “Banking Crises.”
for example, investment banks—to use In The New Palgrave Dictionary of Economics. 2nd ed.
too much debt and extended too much
Edited by Steven N. Durlauf and Lawrence E. Blume.
credit, thus accumulating an excessive amount off-balance-sheet vehicles to hide the risks of Basingstoke, Hampshire, United Kingdom:
of risk. In our opinion, this was a failure in securitization from their financial statements. Palgrave Macmillan.
several dimensions: Cooper, George. 2008. The Origin of Financial Crises:
Therefore, this period saw market inefficiency, Central Banks, Credit Bubbles and the Efficient
3 Regulators allowed such accumulation of risk by inadequate or inconsistent government Market Fallacy. New York: Vintage Books.
allowing excessive leverage. vigilance, and a Wicksell-Fisher-Hayek- Fisher, Irving. 1933. “The Debt-Deflation Theory of
3 Shareholders and boards of directors did not Minsky chain of events leading to excessive Great Depressions.” Econometrica, vol. 1, no. 4
require sound risk management. (October): 337–357.
lending, a bubble, and a crash (see Cooper
3 Market participants underestimated risk. 2008). The crash causes a Keynesian aggregate Greenspan, Alan. 2007. The Age of Turbulence:
3 Academics believed too much in market efficiency Adventures in a New World. New York: Penguin Press.
demand drop with ineffective monetary
and were reluctant to admit the possibility of Hayek, Friedrich A. 1932. “Das Schicksal der
policy because of already low policy interest
market irrationality, even though some had spent Goldwahrung” [The Fate of the Gold Standard].
the previous decade analyzing the technology rates. This is the so-called liquidity trap
Der Deutsche Volkswirt, vol. 6, no. 20.
bubble of the 1990s and the subsequent crash. (see Keynes 1936. For more about the liquidity
Kaplan, Paul D. 2009. “One and a Quarter Centuries
3 Politicians were all too happy to see the economy trap in the current crisis, see Krugman 2008.)
of Stock Market Drawdowns.” Morningstar Alternative
grow at an excessive speed because that was good
Investments Observer (Third Quarter).
in the short run.
What Have We Learned?
Keynes, John Maynard. 1936. The General Theory
3 Financial company CEOs were also quite happy to
To prevent a repeat of the same type of crisis in of Employment, Interest, and Money. New York:
see short-term profits swell, hoping that the
inevitable crash would occur after they had retired
the future, we believe that more comprehensive Harcourt Brace and Company.
and cashed out of the company. regulation of the financial system is necessary. Krugman, Paul. 2008. The Return of Depression
This does not mean that we advocate red Economics and the Crisis of 2008. New York:
tape, but that supervisors must guarantee W.W. Norton.
The events of the residential real estate
markets in the United States and in other transparency and limit leverage. Moreover, this Minsky, Hyman P. 1986. Stabilizing an Unstable
regulation should not only be limited to banks Economy. New Haven, CT: Yale University Press.
countries, such as Spain and Iceland,
summarize the key points of the crisis. Home but also apply to insurance companies, Minsky, Hyman P. 1992. “The Financial Instability
Hypothesis.” The Jerome Levy Economics
prices kept increasing, and people wanted investment banks, other nondepository financial
Institute of Bard College, Working Paper No. 74 (May):
to buy homes, hoping not only to live in them institutions, and their holding companies. http://www.levy.org/pubs/wp74.pdf.
but also to profit from their appreciation in Schumpeter, Joseph A. 1942. Capitalism, Socialism,
value. Mortgage brokers, whose compensation When market participants realized that a crash and Democracy. New York: Harper.
depended on the number and size of mortgages was imminent, they tried to sell all risky
Shiller, Robert J. 2005. “Definition of Irrational
they originated, gave mortgages to as many assets to take refuge in safe investments, such Exuberance.” Accessed at
people as possible, regardless of whether these as short-term government bonds. The leading http://www.irrationalexuberance.com/definition.htm
people could afford the mortgages. Banks, risk models used by most participants did not on 18 November 2009.
in a period of low spreads, were looking for fee consider this possibility. As a result, we believe Smith, Adam. 1776. An Inquiry into the Nature and
income and looked for mortgages to be that risk models must consider scenarios of Causes of the Wealth of Nations. London: Printed for
W. Strahan and T. Cadell.
securitized and sold to investors. Investors, sudden flight to quality, and financial analysts
frustrated by otherwise low yields, were eagerMorningstar Institutional Perspective | October 2009
Déjà vu Around the Word
“We seem to have a once-in-a lifetime crisis every three or four years.”
--Leslie Rahl, found of Capital Market Risk Advisors1
by Paul D. Kaplan, Ph.D., CFA Exhibit 1: British Record: Disaster, Crisis, Recovers, & Growth
10.0
Vice President, Quantitative Research
What started as a mortgage crisis in the United States quickly spread to 1.0
69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 01 03 05 07
nearly every corner of the financial system when Lehman Brothers
collapsed, Merrill Lynch sold itself to Bank of America, and AIG became
strapped for cash—all in a single weekend. These and the events that
followed shook investor confidence to the core. Stock markets around the
world plummeted as exemplified by the FTSE 100 falling 65% from
0.1
September to March. Growth of £1 invested in the MSCI UK Gross Return Index, Inflation adjusted, January 1970 − May 2009
Source: Morningstar EnCorr, MSCI Barra, International Monetary Fund
As the markets for many assets became illiquid, and credit dried up for
Exhibit 2: Largest Peak-to-Trough Declines for the U.K.
almost everyone who needed it, the Bank of England, the U.S. Federal
Peak Trough Decline Recovery
Reserve, the U.S. Treasury, and their counterparts around the world took
April 1972 November 1974 73.81% January 1984
dramatic steps to restore liquidity to asset markets, stimulate lenders to December 1999 January 2003 44.91% April 2007
make loans again, and shore up investor confidence in equity markets in October 2007 February 2009 40.99% To Be Determined
an attempt to avoid a deep global recession. Political and fiscal policy September 1987 November 1987 34.07% November 1992
leaders here in the colonies helped sell their $700 billion bailout package December 1969 May 1970 20.38% May 1971
Month-end inflation-adjusted results as of May 2009 since 1969
last fall as an extraordinary remedy for a “once-in-a-century event.” This Source: Morningstar EnCorr, MSCI Barra, International Monetary Fund
was echoed in November by Henry Paulson, the former U.S. Secretary of
the Treasury, who said the meltdown was a “once- or twice-in-a-100-year Looking at the prosperous island nation at the other side of Eurasia, the
event” and former Federal Reserve Chairman Alan Greenspan who story is even more frightening. Exhibit 3 shows that over the same nearly
characterized the crisis as a “once-in-a-century credit tsunami.” 40-year period, the Japanese stock market is still in its second extended
period of decline; and this one began nearly 20 years ago!
There's little doubt that aspects of this crisis are unique and that the
economy is facing its hardest challenge since the Great Depression, but Exhibit 3: The Japanese Record: Lightening Can Strike Twice
are severe economic crises the rare events Paulson, Greenspan, et al, 10.0
have suggested? A study of capital market history around the world
suggests no, and perhaps nowhere more clearly than in the United
Kingdom. While Americans think of the greatest decline in stock market
history as occurring during the 1930s, for British investors, the worst
decline was in the 1970s. After taking into account the impact of inflation 1.0
and even after reinvesting all dividends, the British stock market fell 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 01 03 05 07
almost 74 percent from April 1972 to November 1972 and took nearly a
decade to recover to its previous level.2
Exhibit 1 illustrates the inflation-adjusted growth of £1 invested at the end
0.1
of 1969 in the MSCI UK Gross Return Index.3 While overall, this Growth of ¥1 invested in the MSCI Japan Gross Return Index, Inflation adjusted, January 1970 − May 2009
Source: Morningstar EnCorr, MSCI Barra, International Monetary Fund
investment would have grown to the equivalent of 5.6 times in purchasing
power by the end of May 2009, the record is peppered with several long
and severe declines. Exhibit 2 lists the worst of these declines.Déjas Vu Around the World Morningstar Institutional Perspective | September 2009
Furthermore, the capital market histories of the United Kingdom and doctorial dissertation, Fama applied Mandelbrot's model to stock prices
Japan are not unique. Exhibit 4 depicts the largest inflation-adjusted and obtained promising results.5 Until recently, however, the work of
declines in eight industrialized countries (including the U.K. and Japan) Mandelbrot and Fama had been largely ignored.6
over the past four decades. All of the largest markets suffered a major
decline over the period, which clearly illustrates that level of stock risk is Exhibit 5: Cracks in the Bell Curve – U.K.
high indeed. Historical Frequency (Months)
Lognormal Historical
128
64
32
Exhibit 4: Largest Peak-to-Trough Declines in Eight Countries Since 1969 16
8
Country Peak Trough Decline Recovery
4
Spain April 1973 April 1980 85.36% December 1996 2
1
Italy January 1970 December 1977 82.58% March 1986 Jan-75
U.K. April 1972 November 1974 73.81% January 1984
Japan December 1989 April 2003 70.33% To Be Determined
Germany February 2000 March 2003 69.44% To Be Determined
France August 2000 March 2003 60.52% To Be Determined
Canada February 1980 June 1982 51.38% March 1986 Monthly Return
U.S. December 1999 February 2009 54.84% To Be Determined
-28% -18% -8% 2% 12% 22% 32% 42% 52%
Month-end results as of May 2009 in inflation-adjusted local currency
Source: Morningstar EnCorr, MSCI Barra, International Monetary Fund Monthly inflation-adjusted returns on the MSCI UK Gross Return index: Jan 1926−May 2009
Source: Morningstar EnCorr, MSCI Barra, and International Monetary Fund
Modeling Risk: The Standard Model
In his dissertation, Fama assumed that the logarithm of stock returns
With large prolonged declines occurring with such frequency, you’d think
followed a fat-tailed distribution called a “stable Paretian distribution,” or
that the standard risk models investors use to make their asset-allocation
stable distribution.7 Hence, we refer to the resulting distribution of returns
decisions would assign a significant probability that these events will
as a "log-stable distribution."
occur. Think again. To see why, we need to look at how these models
were formed.
Exhibit 6 adds the best-fitting log-stable distribution curve to Exhibit 5.
While not perfect, the log-stable model fits the historical distribution much
To help make sense of the highly complex capital markets, financial
closer than the lognormal both at the center and the tails.
economists in 1960s and 1970s developed a set of mathematical models
of the markets. The best known of these models are the Capital Asset
Pricing Model (CAPM) of expected returns and the Black-Scholes Option Exhibit 6: Modeling Fat Tails – U.K.
Historical Frequency (Months) Log-Stable Lognormal Historical
Pricing Model. Their creators won the Nobel Prize in economics for their 128
64
ground-breaking work. Each of these models is built on the assumption 32
that the statistical distribution of market returns follows a normal, or bell- 16
8
shaped, distribution.4 And even though the historical data tells a different 4
story, these models are firmly entrenched throughout the investment 2
1
profession. Jan-75
An Alternative Approach: Log-Stable Distributions
Exhibit 5 shows the distribution of monthly real total returns for the UK
stock market from January 1970 through May 2009 along with the Monthly Return
lognormal distribution curve that best fits the data. (The chart is drawn -28% -18% -8% 2% 12% 22% 32% 42% 52%
Monthly inflation-adjusted returns on the MSCI UK Gross Return index: Jan 1926−May 2009
using a logarithmic scale to emphasis the tails of the distributions.) While Source: Morningstar EnCorr, MSCI Barra, and International Monetary Fund
in most months, the historical returns closely follow the curve, there are
several months that have returns that fall far to the right or left of the
Risk Measures
lognormal curve. It is these outliers in the tails that constitute both the
Our analysis of stock market drawdowns and return distributions strongly
opportunities and the risks of equity investing. This phenomenon is not
suggests that summarizing risk with standard deviation omits much of the
unique to the UK market; rather, it is typical of equity markets throughout
story. We expect to see modeling tools for advisors come to market in the
the world.
near future that can account for large, prolonged drawdowns and fat tails.
In the early 1960s, Benoit Mandelbrot, a mathematician teaching
One modeling approach that is currently used by some institutional money
economics at the University of Chicago, was advising a doctoral student
managers and risk analysts is to use fat-tailed models to develop
named Eugene Fama. Mandelbrot had developed a statistical model for
measures of Value at Risk (VaR) and Expected Shortfall.8 VaR describes
percentage changes in the price of cotton that had “fat tails.” That is, the
the left tail in terms of how much capital can be lost over a given period of
model assigned nontrivial probabilities to large percentage changes. In his
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